Algorithm 537: Characteristic Values of Mathieu's Differential Equation [S22]
ACM Transactions on Mathematical Software (TOMS)
Communications of the ACM
Algorithms for the computation of all Mathieu functions of integer orders
ACM Transactions on Mathematical Software (TOMS)
Algorithm 804: subroutines for the computation of Mathieu functions of integer orders
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
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Two algorithms for calculating the eigenvalues and solutions of Mathieu's differential equation for noninteger order are described. In the first algorithm, Leeb's method is generalized, expanding the Mathieu equation in Fourier series and diagonalizing the symmetric tridiagonal matrix that results. Numerical testing was used to parameterize the minimum matrix dimension that must be used to achieve accuracy in the eigenvalue of one part in 1012. This method returns a set of eigenvalues below a given order and their associated solutions simultaneously. A second algorithm is presented which uses approximations to the eigenvalues (Taylor series and asymptotic expansions) and then iteratively corrects the approximations using Newton's method until the corrections are less than a given tolerance. A backward recursion of the continued fraction expansion is used. The second algorithm is faster and is optimized to obtain accuracy of one part in 1014, but has only been implemented for orders less than 10.5.